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Ma, Weimin; Sun, Bingzhen

Probabilistic rough set over two universes and rough entropy. (English) Zbl 1246.68234

Int. J. Approx. Reasoning 53, No. 4, 608-619 (2012).
Summary: We discuss the properties of the probabilistic rough set over two universes in detail. We present the parameter dependence or the continuous of the lower and upper approximations on parameters for probabilistic rough set over two universes. We also investigate some properties of the uncertainty measure, i.e., the rough degree and the precision, for probabilistic rough set over two universes. Meanwhile, we point out the limitation of the uncertainty measure for the traditional method and then define the general Shannon entropy of covering-based on universe. Then we discuss the uncertainty measure of the knowledge granularity and rough entropy for probabilistic rough set over two universes by the proposed concept. Finally, the validity of the methods and conclusions is tested by a numerical example.

Cited in 33 Documents

MSC:

68T37 Reasoning under uncertainty in the context of artificial intelligence
94A17 Measures of information, entropy

Keywords:

rough set; probabilistic approximation space over two universes; general Shannon entropy
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\textit{W. Ma} and \textit{B. Sun}, Int. J. Approx. Reasoning 53, No. 4, 608--619 (2012; Zbl 1246.68234)
Full Text: DOI

References:

[1] Pawlak, Z., Rough sets, Internal Journal Information Science, 11, 5, 341-356 (1982) · Zbl 0501.68053
[2] Pawlak, Z., Rough Sets: Theoretical Aspects of Reasoning About Data (1991), Springer · Zbl 0758.68054
[3] Pawlak, Z.; Skowron, A., Rudiments of rough sets, Information Sciences, 177, 1, 3-27 (2007) · Zbl 1142.68549
[4] Lingras, P. J.; Yao, Y. Y., Data mining using extensions of the rough set model, Journal of the American Society for Information Science, 49, 5, 415-422 (1998) · Zbl 0923.04007
[5] Yamaguchi, D., Attribute dependency functions considering data efficiency, International Journal of Approximate Reasoning, 51, 1, 89-98 (2009)
[6] Li, T. R.; Ruan, D.; Geert, W.; Song, J.; Xu, Y., A rough sets based characteristic relation approach for dynamic attribute generalization in data mining, Knowledge-Based Systems, 20, 5, 485-494 (2007)
[7] Mac Parthalain, N.; Shen, Q., Exploring the boundary region of tolerance rough sets for feature selection, Pattern Recognition, 42, 5, 655-667 (2009) · Zbl 1162.68625
[8] Skowron, A.; Stepaniuk, J., Tolerance approximation space, Fundanation Information, 27, 245-253 (1996) · Zbl 0868.68103
[9] Zhu, W., Generalized rough sets based on relations, Information Sciences, 177, 4997-5011 (2007) · Zbl 1129.68088
[10] Kryszkiewicz, M., Comparative study of alternative type of knowledge reduction in inconsistent systems, International Journal of Intelligent Systems, 16, 105-120 (2001) · Zbl 0969.68146
[11] Yao, Y. Y., Constructive and algebraic methods of the theory of rough set, Information Sciences, 109, 21-47 (1998) · Zbl 0934.03071
[12] Radzikowska, A. M.; Kerre, E. E., A comparative study of fuzzy rough sets, Fuzzy Sets and Systems, 126, 137-155 (2002) · Zbl 1004.03043
[13] Sun, B. Z.; Gong, Z. T.; Chen, D. G., Fuzzy rough set theory for the interval-valued fuzzy information systems, Information Sciences, 178, 13, 2794-2815 (2008) · Zbl 1149.68434
[14] Yao, Y. Y., A comparative study of fuzzy sets and rough sets, Journal of Information Science, 109, 227-242 (1998) · Zbl 0932.03064
[15] Li, T. J.; Leung, Y.; Zhang, W. X., Generalized fuzzy rough approximation operators based on fuzzy coverings, International Journal of Approximate Reasoning, 48, 3, 836-856 (2008) · Zbl 1186.68464
[16] Wu, W. Z.; Zhang, W. X., Generalized fuzzy rough sets, Information Science, 15, 263-282 (2003) · Zbl 1019.03037
[17] Leung, Y.; Fischer, M.; Wu, W.; Mi, J., A rough set approach for the discovery of classification rules in interval-valued information systems, International Journal of Approximate Reasoning, 47, 2, 233-246 (2008) · Zbl 1184.68409
[18] Yao, Y. Y., Two views of the theory of rough sets in finite universes, International Journal of Approximate Reasoning, 15, 4, 291-318 (1996) · Zbl 0935.03063
[19] Slowinski, R.; Vanderpooten, D., A generalized definition of rough approximations based on similarity, IEEE Transactions on Knowledge and Data Engineering, 12, 331-336 (2000)
[20] Greco, S.; Matarazzo, B.; Slowinski, R., Rough approximation of a preference relation by dominance relations, European Journal of Operational Research, 117, 1, 63-83 (1999) · Zbl 0998.90044
[21] Pei, Z.; Pei, D. W.; Li, Z., Topology vs generalized rough sets, International Journal of Approximate Reasoning, 52, 2, 231-239 (2011) · Zbl 1232.03044
[22] Yao, Y., Relational interpretations of neighborhood operators and rough set approximation operators, Information Sciences, 111, 1-4, 239-259 (1998) · Zbl 0949.68144
[23] Ziarko, W., Variable precision rough set model, Journal of Computer and System Sciences, 46, 39-59 (1993) · Zbl 0764.68162
[24] Gong, Z. T.; Sun, B. Z., Variable precision rough set model based on general relations, Journal of Lanzhou University (Natural Sciences), 41, 6, 110-114 (2005) · Zbl 1115.68516
[25] Beynon, M., Reducts within the variable precision rough sets model: a further investigation, European Journal of Operational Research, 134, 3, 592-605 (2001) · Zbl 0984.90018
[26] Salvatore, G.; Matarazzo, B.; Sowiski, R., Parameterized rough set model using rough membership and Bayesian confirmation measures, International Journal of Approximate Reasoning, 49, 2, 285-300 (2008) · Zbl 1191.68678
[27] Sun, B. Z.; Gong, Z. T., Variable precision probabilistic rough set model, Journal of Northwest Normal University (Natural Science), 41, 4, 23-26 (2005) · Zbl 1087.68669
[28] Shi, Y.; Yao, L. M.; Xu, J. P., A probability maximization model based on rough approximation and its application to the inventory problem, International Journal of Approximate Reasoning, 52, 2, 261-280 (2011) · Zbl 1222.90056
[29] Huynh, V. N.; Nakamori, Y., A roughness measure for fuzzy sets, Information Sciences, 173, 255-275 (2005) · Zbl 1074.03024
[30] Dubois, D.; Prade, H., Rough fuzzy sets and fuzzy rough sets, International Journal of General Systems, 17, 191-209 (1990) · Zbl 0715.04006
[31] Banerjee, M.; Pal, S. K., Roughness of a fuzzy set, Information Sciences, 93, 235-246 (1996) · Zbl 0879.04004
[32] Wu, W. Z.; Leung, Y.; Mi, J. S., On characterizations of \((I, T)\)-fuzzy rough approximation operators, Fuzzy Sets and Systems, 154, 76-102 (2005) · Zbl 1074.03027
[33] Liu, G. L., Axiomatic systems for rough sets and fuzzy rough sets, International Journal of Approximate Reasoning, 48, 3, 857-867 (2008) · Zbl 1189.03056
[34] Shen, Y. H.; Wang, F. X., Rough approximations of vague sets in fuzzy approximation space, International Journal of Approximate Reasoning, 52, 2, 281-296 (2011) · Zbl 1232.03045
[35] Duntsch, I.; Gediga, G., Uncertainty measures of rough set prediction, Artificial Intelligence, 106, 1, 109-137 (1998) · Zbl 0909.68040
[36] Liang, J. Y.; Li, D. Y., Knowledge Acquirment and Uncertainty Measurement for Information Systems (2005), Science Press: Science Press Beijing, pp. 39-46
[37] Yao, Y. Y., Probabilistic rough set approximations, International Journal of Approximate Reasoning, 49, 2, 255-271 (2008) · Zbl 1191.68702
[38] Ziarko, W., Probabilistic approach to rough sets, International Journal of Approximate Reasoning, 49, 2, 272-284 (2008) · Zbl 1191.68705
[39] Wong, S. K.M.; Ziarko, W., Comparison of the probabilistic approximate classification and the fuzzy set model, Fuzzy Sets and Systems, 21, 357-362 (1987) · Zbl 0618.60002
[40] Slezak, D.; Ziarko, W., The investigation of the Bayesian rough set model, International Journal of Approximate Reasoning, 40, 1-2, 81-91 (2005) · Zbl 1099.68089
[41] Yao, J. T.; Yao, Y. Y.; Ziarko, W., Probabilistic rough sets: approximations, decision-makings, and applications, International Journal of Approximate Reasoning, 49, 2, 253-254 (2008)
[43] Wong, S. K.M.; Wang, L. S.; Yao, Y. Y., On modeling uncertainty with interval structure, Computer and Intelligence, 11, 406-426 (1993)
[44] Pei, D. W.; Xu, Z. B., Rough set models on two universes, International Journal of General Systems, 33, 5, 569-581 (2004) · Zbl 1154.93378
[45] Zhang, W. X.; Wu, W. Z., Rough set models based on random sets (I), Journal of Xi’an Jiaotong University, 12, 75-79 (2000)
[46] Zhang, W. X.; Wu, W. Z., Rough set models based on random sets (II), Journal of Xi’an Jiaotong University, 35, 4, 425-429 (2000)
[47] Sun, B. Z.; Ma, W. M., Fuzzy rough set model on two different universes and its application, Applied Mathematical Modelling, 35, 4, 1798-1809 (2011) · Zbl 1217.03041
[48] Zhang, H. Y.; Zhang, W. X.; Wu, W. Z., On characterization of generalized interval-valued fuzzy rough sets on two universes of discourse, International Journal of Approximate Reasoning, 51, 1, 56-70 (2009) · Zbl 1209.68552
[49] Li, T. J., Rough approximation operators on two universes of discourse and their fuzzy extensions, Fuzzy Sets and Systems, 159, 033-3050 (2008) · Zbl 1176.03036
[50] Shen, Y. H.; Wand, F. X., Variable precision rough set model over two universes and its properties, Soft Computing, 15, 3, 557-567 (2011) · Zbl 1237.68216
[52] Yan, R. X.; Zheng, J. G.; Liu, J. L.; Zhai, Y. M., Research on the model of rough set over dual-universes, Knowledge-Based Systems, 23, 817-822 (2010)
[54] Shannon, C. E., The mathematical theory of communication, The Bell System Technical Journal, 12, 3-4, 373-423 (1948) · Zbl 1154.94303
[55] Bianucci, D.; Cattaneo, Gia, Information entropy and granulation co-entropy of partitions and coverings: a summary, Transactions on Rough Sets X LNCS, 5656, 15-66 (2009) · Zbl 1248.94038
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